The F-rational signature and drops in the Hilbert–Kunz multiplicity

Hochster, Melvin; Yao, Yongwei

doi:10.2140/ant.2022.16.1777

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The F-rational signature and drops in the Hilbert–Kunz multiplicity

Melvin Hochster and Yongwei Yao

Vol. 16 (2022), No. 8, 1777–1809

DOI: 10.2140/ant.2022.16.1777

Abstract

Let $(R, 𝔪)$ be a Noetherian local ring of prime characteristic $p$ . We define the F-rational signature of $R$ , denoted by $r (R)$ , as the infimum, taken over pairs of ideals $I ⊊ J$ such that $I$ is generated by a system of parameters and $J$ is a strictly larger ideal, of the drops $e_{HK} (I, R)) - e_{HK} (J, R)$ in the Hilbert–Kunz multiplicity. If $R$ is excellent, then $R$ is F-rational if and only if $r (R) > 0$ . The proof of this fact depends on the following result in the sequel: Given an $𝔪$ -primary ideal $I$ in $R$ , there exists a positive $δ_{I} \in ℝ^{+}$ such that, for any ideal $J ⊋ I$ , $e_{HK} (I, R) - e_{HK} (J, R)$ is either $0$ or at least $δ_{I}$ . We study how the F-rational signature behaves under deformation, flat local ring extension, and localization.

Keywords

F-rational signature, F-signature, Hilbert–Kunz multiplicity, Frobenius, Cohen–Macaulay, Gorenstein, regular

Mathematical Subject Classification 2010

Primary: 13A35

Secondary: 13C13, 13H10

Milestones

Received: 19 September 2017

Revised: 8 August 2021

Accepted: 22 November 2021

Published: 29 November 2022

Authors

Melvin Hochster
Department of Mathematics University of Michigan Ann Arbor MI United States

Yongwei Yao
Department of Mathemaics and Statistics Georgia State University Atlanta GA United States

Vol. 16, No. 8, 2022